Two-dimensional area minimizing integral currents are classical minimal surfaces

Author:
Sheldon Xu-Dong Chang

Journal:
J. Amer. Math. Soc. **1** (1988), 699-778

MSC:
Primary 49F20; Secondary 49F10, 49F22, 58E12, 58E15

DOI:
https://doi.org/10.1090/S0894-0347-1988-0946554-0

MathSciNet review:
946554

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Abstract: Geometric measure theory guarantees the existence of area minimizing integral currents spanning a given boundary or representing a given integral homology class on a compact Riemannian manifold. We study the regularity of such generalized surfaces. We prove that in case the dimension of the area minimizing integral currents is two, then they are classical minimal surfaces. Among the consequences of this regularity result, we know now that any two dimensional integral homology class on a compact Riemannian manifold can be represented by a finite integral linear combination of classical closed minimal surfaces that have only finitely many intersection points.

The result is proved by using the theory of multiple-valued functions developed by F. Almgren in [A]. We extend many important estimates in his paper and extend his construction of center manifolds. We use the branched center manifolds and lowest order term in the multiple-valued functions approximating the area minimizing currents to construct two sequences of branched surfaces near an interior singular point to separate the nearby singularity gradually. The analysis developed in this paper enables us to conclude the generalized surface must coincide with one of the branched surfaces.

**[WA]**William K. Allard,*On the first variation of a varifold*, Ann. of Math. (2)**95**(1972), 417–491. MR**0307015**, https://doi.org/10.2307/1970868**[A]**F. J. Almgren,*valued functions minimizing Dirichlet's integral and the regularity of area minimizing rectifiable currents up to codimension two*, preprint.**[C]**S. Chang,*The structure of the singularity of multiple-valued functions minimizing Dirichlet integral over two dimensional domain*, preprint.**[F]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[FM]**Frank Morgan,*On the singular structure of three-dimensional, area-minimizing surfaces*, Trans. Amer. Math. Soc.**276**(1983), no. 1, 137–143. MR**684498**, https://doi.org/10.1090/S0002-9947-1983-0684498-4**[LS]**Leon Simon,*Lectures on geometric measure theory*, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR**756417****[LS1]**Leon Simon,*Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems*, Ann. of Math. (2)**118**(1983), no. 3, 525–571. MR**727703**, https://doi.org/10.2307/2006981**[S]**M. Spivak,*A comprehensive introduction to differential geometry*, vol. 4, Chapter 9, Addendum 1.*Isothermal coordinates*, Publish or Perish, Boston, 1975.**[BW]**Brian White,*Tangent cones to two-dimensional area-minimizing integral currents are unique*, Duke Math. J.**50**(1983), no. 1, 143–160. MR**700134**, https://doi.org/10.1215/S0012-7094-83-05005-6**[Y]**S. T. Yau,*Survey on partial differential equations in differential geometry*, Seminar on Differential Geometry (S. T. Yau, ed.), vol. 102, Ann. of Math. Studies.**[GMT]***Geometric measure theory and the calculus of variations*, Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1985.

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DOI:
https://doi.org/10.1090/S0894-0347-1988-0946554-0

Article copyright:
© Copyright 1988
American Mathematical Society